2015/08/27 by Jayne, Rebecca L., Misra, Kailash C.
#05E10 #17B10 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1508.06930
For ℓ ≥ 1 and k ≥ 2, we consider certain admissible sequences of k-1 lattice paths in a colored ℓ × ℓ square. We show that the number of such admissible sequences of lattice paths is given by the sum of squares of the number of standard Young tableaux of partitions of ℓ with height ≤ k, which is also the number of (k+1)k⋯21-avoiding permutations of \1, 2, …, ℓ\. Finally, we apply this result to the representation theory of the affine Lie algebra \widehatsl(n) and show that this quantity gives the multiplicity of certain maximal dominant weights in the irreducible module V(kΛ0).